Let \(\mathbf {f}\)  be an eigenvector corresponding to
\(q(G)\) , i.e., \(Q(G)\mathbf {f}= q(G)\mathbf {f}\) .
By the celebrated Perron–Frobenius theorem (see [1]}
or [2]}),
we may assume that \(f_{v}>0\)  for each \(v\in V(G)\) 
when \(G\)  is connected.
It is easy to see from the eigen-equation that
for any \(u,v\in V(G)\) ,
\((q(G)- d(u)) f_u =\sum _{w\in N(u)} f_w, ~~\text{and}~~(q(G)- d(v)) f_v =\sum _{z\in N(v)} f_z.\) 
